9.8 APPLICATIONS OF TAYLOR SERIES EXPLORATORY EXERCISES. Using Taylor Polynomials to Approximate a Sine Value EXAMPLE 8.1

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1 9-75 SECTION Applications of Taylor Series 677 and f 0) miles/min 3. Predict the location of the plane at time t min. 5. Suppose that an astronaut is at 0, 0) and the moon is represented by a circle of radius centered at 0, 5). The astronaut s capsule follows a path y f ) with current position f 0) 0, slope f 0) /5, concavity f 0) /0, f 0) /5, f 4) 0) /5 and f 5) 0) /50. Graph a Taylor polynomial approimation of f ). Based on your current information, do you advise the astronaut to change paths? How confident are you in the accuracy of your approimation? 53. Find the Taylor series for e about a general center c. 54. Find the Taylor series for about a general center c a. Eercises involve the binomial epansion. 55. Show that the Maclaurin series for + ) r is rr ) r + ) +, for any constant r.! 56. Simplify the series in eercise 55 for r ; r 3; r is a positive integer. 57. Use the result of eercise 55 to write out the Maclaurin series for f ) Use the result of eercise 55 to write out the Maclaurin series for f ) + ) 3/. 59. Find the Maclaurin series of f ) cosh and f ) sinh. Compare to the Maclaurin series of cos and sin. 60. Use the Maclaurin series for tan and the result of eercise 59 to conjecture the Maclaurin series for tanh. EXPLORATORY EXERCISES. Almost all of our series results apply to series of comple numbers. Defining i, show that i, i 3 i, i 4 and so on. Replacing with i in the Maclaurin series for e, separate terms containing i from those that don t contain i after the simplifications indicated above) and derive Euler s formula: e i cos + i sin.. Using the technique of eercise, show that cosi) cosh and sini) i sinh. That is, the trig functions and their hyperbolic counterparts are closely related as functions of comple variables. 3. The method used in eamples 7.3, 7.5, 7.6 and 7.7 does not require us to actually find R n ), but to approimate it with a worst-case bound. Often this approimation is fairly close to R n ), but this is not always true. As an etreme eample of this, show that the bound on R n ) for f ) ln about c see eercise 3) increases without bound for 0 < <. Eplain why this does not necessarily mean that the actual error increases without bound. In fact, R n ) 0 for 0 < < but we must show this using some other method. Use integration of an appropriate power series to show that ) + ) converges to ln for 0 < <. 4. Verify numerically that if a is close to π, the sequence a n+ a n + sin a n converges to π. In other words, if a n is an approimation of π, then a n + sin a n is a better approimation.) To prove this, find the Taylor series for sin about c π. Use this to show that if π<a n < π, then π<a n+ < a n. Similarly, show that if 0 < a n <π, then a n < a n+ <π. 9.8 APPLICATIONS OF TAYLOR SERIES In section 9.7, we developed the concept of a Taylor series epansion and gave many illustrations of how to compute these. In this section, we epand on our earlier presentation, by giving a few eamples of how Taylor series are used to approimate the values of transcendental functions, evaluate limits and integrals and define important new functions. These represent but a small sampling of the important applications of Taylor series. First, consider how calculators and computers might calculate values of transcendental functions, such as sin.34567). We illustrate this in eample 8.. EXAMPLE 8. Using Taylor Polynomials to Approimate a Sine Value Use a Taylor series to approimate sin.34567) accurate to within 0.

2 678 CHAPTER 9.. Infinite Series 9-76 Solution In section 9.7, we left it as an eercise to show that the Taylor series epansion for f ) sin about 0is sin ) + )! + 3! 3 + 5! 5 7! 7 +, where the interval of convergence is, ). Notice that if we tae.34567, the series representation of sin is sin ) + )!.34567)+, which is an alternating series. We can use a partial sum of this series to approimate the desired value, but how many terms will we need for the desired accuracy? Recall that for alternating series, the error in a partial sum is bounded by the absolute value of the first neglected term. Note that you could also use the remainder term from Taylor s Theorem to bound the error.) To ensure that the error is less than 0, we must find an integer such that < 0. By trial and error, we find that + )! < 0, 7! so that 8 will do. This says that the first neglected term corresponds to 8 and so, we compute the partial sum 7 ) sin )!.34567) ! ! ! ! Chec your calculator or computer to verify that this matches your calculator s estimate. In eample 8., while we produced an approimation with the desired accuracy, we did not do this in the most efficient fashion, as we simply grabbed the most handy Taylor series epansion of f ) sin. You should try to resist the impulse to automatically use the Taylor series epansion about 0 i.e., the Maclaurin series), rather than maing a more efficient choice. We illustrate this in eample 8.. EXAMPLE 8. Choosing a More Appropriate Taylor Series Epansion Repeat eample 8., but this time, mae a more appropriate choice of the Taylor series. Solution Recall that Taylor series converge much faster close to the point about which you epand, than they do far away. Given this and the fact that we now the eact value of sin at only a few points, you should quicly recognize that a series epanded about π.57 is a better choice for computing sin than one epanded about 0. Another reasonable choice is the Taylor series epansion about π 3.) In eample 7.5, recall that we had found that ) sin π ) π ) + π ) 4, )! 4!

3 9-77 SECTION Applications of Taylor Series 679 where the interval of convergence is, ). Taing gives us ) sin π ) )! π ) π ) 4, 4! which is again an alternating series. Using the remainder term from Taylor s Theorem to bound the error, we have that R n.34567) f n+) z) n + )! π n π n+. n + )! Note that we get the same error bound if we use the error bound for an alternating series.) By trial and error, you can find that π n+ < 0 n + )! for n 4, so that an approimation with the required degree of accuracy is sin ) 6! )! π ) π π ) + 4! ) 6 + 8! π ) π Compare this result to eample 8., where we needed many more terms of the Taylor series to obtain the same degree of accuracy. We can also use Taylor series to quicly conjecture the value of difficult limits. Be careful, though: the theory of when these conjectures are guaranteed to be correct is beyond the level of this tet. However, we can certainly obtain helpful hints about certain limits. EXAMPLE 8.3 Using Taylor Polynomials to Conjecture the Value of a Limit sin 3 3 Use Taylor series to conjecture lim. 0 9 Solution Again recall that the Maclaurin series for sin is sin ) + )! + 3! 3 + 5! 5 7! 7 +, where the interval of convergence is, ). Substituting 3 for gives us sin 3 ) + )! 3 ) ! + 5 5!. ) 8

4 680 CHAPTER 9.. Infinite Series 9-78 This gives us sin and so, we conjecture that 3 9 3! + 5 ) 5! 3 9 3! + 6 sin 3 3 lim 0 9 3! 6. 5! + You can verify that this limit is correct using l Hôpital s Rule three times, simplifying each time). Since Taylor polynomials are used to approimate functions on a given interval and since polynomials are easy to integrate, we can use a Taylor polynomial to obtain an approimation of a definite integral. It turns out that such an approimation is often better than that obtained from the numerical methods developed in section 4.7. We illustrate this in eample 8.4. EXAMPLE 8.4 Using Taylor Series to Approimate a Definite Integral Use a Taylor polynomial with n 8 to approimate cos ) d. Solution Since we do not now an antiderivative of cos ), we must rely on a numerical approimation of the integral. Since we are integrating on the interval, ), a Maclaurin series epansion i.e., a Taylor series epansion about 0) is a good choice. We have ) cos )! + 4! 4 6! 6 +, which converges on all of, ). Replacing by gives us cos ) ) )! ! 8 6! +, so that cos ) 4 + 4! 8. This leads us to the approimation cos ) d 4 + 4! ) 8 d ) Our CAS gives us cos ) d , so our approimation appears to be very accurate. You might reasonably argue that we don t need Taylor series to obtain approimations lie those in eample 8.4, as you could always use other, simpler numerical methods lie Simpson s Rule to do the job. That s often true, but just try to use Simpson s Rule on the integral in eample 8.5.

5 9-79 SECTION Applications of Taylor Series 68 EXAMPLE 8.5 Using Taylor Series to Approimate the Value of an Integral Use a Taylor polynomial with n 5 to approimate sin Solution Note that you do not now an antiderivative of sin. Further, while the integrand is discontinuous at 0, this does not need to be treated as an improper sin integral, since lim. This says that the integrand has a removable 0 discontinuity at 0.) From the first few terms of the Maclaurin series for f ) sin, we have the Taylor polynomial approimation so that Consequently, Our CAS gives us sin sin 3 3! + 5 5!, sin 3! + 4 5!. d ) d ) ) d sin d.896, so our approimation is quite good. On the other hand, if you try to apply Simpson s Rule or Trapezoidal Rule, the algorithm will not wor, as they will attempt to evaluate sin at 0. While you have now calculated Taylor series epansions of many familiar functions, many other functions are actually defined by a power series. These include many functions in the very important class of special functions that frequently arise in physics and engineering applications. One important family of special functions are the Bessel functions, which arise in the study of fluid mechanics, acoustics, wave propagation and other areas of applied mathematics. The Bessel function of order p is defined by the power series J p ) ) +p +p! + p)!, 8.) for nonnegative integers p. Bessel functions arise in the solution of the differential equation y + y + p )y 0. In eamples 8.6 and 8.7, we eplore several interesting properties of Bessel functions. EXAMPLE 8.6 The Radius of Convergence of a Bessel Function Find the radius of convergence for the series defining the Bessel function J 0 ). )

6 68 CHAPTER 9.. Infinite Series 9-80 Solution From equation 8.) with p 0, we have J 0 ) ). The Ratio!) Test gives us lim a + a lim +!) + [ + )!] lim 4 + ) 0 <, for all. The series then converges absolutely for all and so, the radius of convergence is. In eample 8.7, we eplore an interesting relationship between the zeros of two Bessel functions y y J 0 ) y J ) FIGURE 9.43 y J 0 ) and y J ) EXAMPLE 8.7 The Zeros of Bessel Functions Verify graphically that on the interval [0, 0], the zeros of J 0 ) and J ) alternate. Solution Unless you have a CAS with these Bessel functions available as built-in functions, you will need to graph partial sums of the defining series: n ) n ) + J 0 ) and J!) ) +! + )!. Before graphing these, you must first determine how large n should be in order to produce a reasonable graph. Notice that for each fied > 0, both of the defining series are alternating series. Consequently, the error in using a partial sum to approimate the function is bounded by the first neglected term. That is, and J ) J 0) n )!) n+ n+ [n + )!] n ) + +! + )! n+3 n+3 n + )!n + )!, with the maimum error in each occurring at 0. Notice that for n, we have that J ) 0)!) )+ )+ [ + )!] !) < 0.04 and J ) ) + +! + )! )+3 )+3 + )! + )! !)4!) < So, in either case, using a partial sum with n results in an approimation that is within 0.04 of the correct value for each in the interval [0, 0]. This is plenty of accuracy for our present purposes. Figure 9.43 shows graphs of partial sums with n for J 0 ) and J ). Notice that J 0) 0 and in the figure, you can clearly see that J 0 ) 0 at about.4, J ) 0 at about 3.9, J 0 ) 0 at about 5.6, J ) 0 at about 7.0 and J 0 ) 0 at about 8.8. From this, it is now apparent that the zeros of J 0 ) and J ) do indeed alternate on the interval [0, 0]. It turns out that the result of eample 8.7 generalizes to any interval of positive numbers and any two Bessel functions of consecutive order. That is, between consecutive zeros of

7 9-8 SECTION Applications of Taylor Series 683 J p ) is a zero of J p+ ) and between consecutive zeros of J p+ ) is a zero of J p ). We eplore this further in the eercises. The Binomial Series You are already familiar with the Binomial Theorem, which states that for any positive integer n, a + b) n a n + na n n n ) b + a n b + +nab n + b n. n ) We often write this as a + b) n n a n b, where we use the shorthand notation ) ) n n, n, 0 ) n n ) to denote the binomial coefficient, defined by ) n nn ) nn ) n + ), for 3.! For the case where a and b, the Binomial Theorem simplifies to n ) + ) n n. Newton discovered that this result could be etended to include values of n other than positive integers. What resulted is a special type of power series nown as the binomial series, which has important applications in statistics and physics. We begin by deriving the Maclaurin series for f ) + ) n, for some constant n 0. Computing derivatives and evaluating these at 0, we have f ) + ) n f 0) f ) n + ) n f 0) n f ) nn ) + ) n f 0) nn ).. f ) ) nn ) n + ) f ) 0) nn ) n + ). We call the resulting Maclaurin series the binomial series, given by f ) 0) + n + nn ) + +nn ) n + )!!! + ) n. From the Ratio Test, we have lim a + a lim nn ) n + )n ) +! + )! nn ) n + ) n lim +, so that the binomial series converges absolutely for < and diverges for >. By showing that the remainder term R ) tends to zero as, we can confirm that the binomial series converges to + ) n for <. We state this formally in Theorem 8.. and

8 684 CHAPTER 9.. Infinite Series 9-8 THEOREM 8. Binomial Series) ) For any real number r, + ) r r, for < <. As seen in the eercises, for some values of the eponent r, the binomial series also converges at one or both of the endpoints ±. EXAMPLE 8.8 Using the Binomial Series Using the binomial series, find a Maclaurin series for f ) + and use it to approimate 7 accurate to within Solution From the binomial series with r,wehave + + ) / ) / , ) ) + ) 3! ) ) for < <. To use this to approimate 7, we first rewrite it in a form involving +, for < <. Observe that we can do this by writing Since is in the interval of convergence, < <, the binomial series gives us 6 [ ) ) + ) 3 5 ) 4 + ] Since this is an alternating series, the error in using the first n terms to approimate the sum is bounded by the first neglected term. So, if we use only the first three terms of the series, the error is bounded by 3 6 6) > Similarly, if we use the first four terms of the series to approimate the sum, the error is bounded by 5 ) < , as desired. So, we can achieve the desired accuracy by summing the first four terms of the series: [ ) ) + ) ] , where this approimation is accurate to within the desired accuracy. EXERCISES 9.8 WRITING EXERCISES. In eample 8., we showed that an epansion about π is more accurate for approimating sin.34567) than an epansion about 0 with the same number of terms. Eplain why an epansion about. would be even more efficient, but is not practical.. Assuming that you don t need to rederive the Maclaurin series for cos, compare the amount of wor done in eample 8.4 to the wor needed to compute a Simpson s Rule approimation with n 6.

9 9-83 SECTION Applications of Taylor Series In equation 8.), we defined the Bessel functions as series. This may seem lie a convoluted way of defining a function, but compare the levels of difficulty doing the following with a Bessel function versus sin : computing f 0), computing f.), evaluating f ), computing f ), computing f ) d and computing f ) d Discuss how you might estimate the error in the approimation of eample 8.4. In eercises 6, use an appropriate Taylor series to approimate the given value, accurate to within 0.. sin.6. sin cos cos e e 0.4 In eercises 7, use a nown Taylor series to conjecture the value of the limit. 7. lim 0 cos 4 8. lim 0 sin 6 ln ) tan 9. lim 0. lim ) 0 3. lim 0 e. lim 0 e In eercises 3 8, use a nown Taylor polynomial with n nonzero terms to estimate the value of the integral sin d, n 3 4. e d, n 5 6. ln d, n 5 8. π π 0 9. Find the radius of convergence of J ). 0. Find the radius of convergence of J ). 0 cos d, n 4 tan d, n 5 e d, n 4. Find the number of terms needed to approimate J ) within 0.04 for in the interval [0, 0].. Show graphically that the zeros of J ) and J ) alternate on the interval 0, 0]. 3. Einstein s theory of relativity states that the mass of an object traveling at velocity v is mv) m 0 / v /c, where m 0 is the rest mass of the object and c is the speed of light. Show m0 ) that m m 0 + v. Use this approimation to estimate c how large v would need to be to increase the mass by 0%. 4. Find the fourth-degree Taylor polynomial epanded about v 0, for mv) in eercise The weight force due to gravity) of an object of mass m and altitude miles above the surface of the earth is w) mgr, where R is the radius of the earth and g is the R + ) acceleration due to gravity. Show that w) mg /R). Estimate how large would need to be to reduce the weight by 0%. 6. Find the second-degree Taylor polynomial for w) in eercise 5. Use it to estimate how large needs to be to reduce the weight by 0%. 7. Based on your answers to eercises 5 and 6, is weight significantly different at a high-altitude location e.g., 7500 ft) compared to sea level? 8. The radius of the earth is up to 300 miles larger at the equator than it is at the poles. Which would have a larger effect on weight, altitude or latitude? In eercises 9 3, use the Maclaurin series epansion tanh The tangential component of the space shuttle s velocity during g reentry is approimately vt) v c tanh t + tanh v ) 0, v c v c where v 0 is the velocity at time 0 and v c is the terminal velocity see Long and Weiss, The American Mathematical Monthly, February 999). If tanh v 0 v c, show that vt) gt + v c. Is this estimate of vt) too large or too small? 30. Show that in eercise 9, vt) v c as t. Use the approimation in eercise 9 to estimate the time needed to reach 90% of the terminal velocity. 3. The downward velocity of a sy diver of mass m is vt) ) g 40mg tanh 40m t. Show that vt) gt g 0m t The velocity of a water wave of length L in water of depth h satisfies the equation v gl πh tanh π L. Show that v gh. In eercises 33 36, use the Binomial Theorem to find the first five terms of the Maclaurin series. 33. f ) 34. f ) f ) f ) + ) 4/5 + 3 In eercises 37 and 38, use the Binomial Theorem to approimate the value to within a) 6 b) a) 3 b) Apply the Binomial Theorem to + 4) 3 and ) 4. Determine the number of nonzero terms in the binomial epansion for any positive integer n. 40. If n) and are positive integers with n >, show that n n!!n )!.

10 686 CHAPTER 9.. Infinite Series Use eercise 33 to find the Maclaurin series for use it to find the Maclaurin series for sin. and 4. Use the Binomial Theorem to find the Maclaurin series for + ) 4/3 and compare this series to that of eercise 34. π sin 43. Use a Taylor polynomial to estimate d accurate to 0 within This value will be used in the net section.) 44. Use a Taylor polynomial to conjecture the value of e + e and then confirm your conjecture using lim 0 l Hôpital s Rule. 45. The energy density of electromagnetic radiation at wavelength λ from a blac body at temperature T degrees Kelvin) is given by Planc s law of blac body radiation: 8πhc f λ), where h is Planc s constant, c is λ 5 e hc/λt ) the speed of light and is Boltzmann s constant. To find the wavelength of pea emission, maimize f λ) by minimizing gλ) λ 5 e hc/λt ). Use a Taylor polynomial for e with n 7 to epand the epression in parentheses and find the critical number of the resulting function. Hint: Use hc 0.04.) Compare this to Wien s law: λ ma Wien s law is T accurate for small λ. Discuss the flaw in our use of Maclaurin series. 46. Use a Taylor polynomial for e to epand the denominator in Planc s law of eercise 45 and show that f λ) 8πT. State λ 4 whether this approimation is better for small or large wavelengths λ. This is nown in physics as the Rayleigh-Jeans law. 47. The power of a reflecting telescope is proportional to the surface [ area S of the parabolic reflector, where S 8π ) d 3/ 3 c 6c + ]. Here, d is the diameter of the parabolic reflector, which has depth with c d 4. ) d 3/ Epand the term 6c + d and show that if is small, 6c then S πd A dis of radius a has a charge of constant density σ. Point P lies at a distance r directly above the dis. The electrical potential at point P is given by V πσ r + a r). Show that for large r, V πa σ. r EXPLORATORY EXERCISES. The Bessel functions and Legendre polynomials are eamples of the so-called special functions. For nonnegative integers n, the Legendre polynomials are defined by [n/] P n ) n ) n )! n )!!n )! n. Here, [n/] is the greatest integer less than or equal to n/ for eample, [/] 0 and [/] ). Show that P 0 ), P ) and P ) 3. Show that for these three functions, P m )P n ) d 0, for m n. This fact, which is true for all Legendre polynomials, is called the orthogonality condition. Orthogonal functions are commonly used to provide simple representations of complicated functions.. Use the) Ratio Test to show that the radius of convergence of n is. a) If n, show that the interval of convergence is, ). b) If n > 0 and n is not an integer, show that the interval of convergence is [, ]. c) If < n < 0, show that the interval of convergence is, ]. 3. Suppose that p is an approimation of π with p π < Eplain why p has at least two digits of accuracy and has a decimal epansion that starts p Use Taylor s Theorem to show that p + sin p has si digits of accuracy. In general, if p has n digits of accuracy, show that p + sin p has 3n digits of accuracy. Compare this to the accuracy of p tan p. 9.9 FOURIER SERIES Many phenomena we encounter in the world around us are periodic in nature. That is, they repeat themselves over and over again. For instance, light, sound, radio waves and -rays are all periodic. For such phenomena, Taylor polynomial approimations have shortcomings. As gets farther away from c the point about which you epanded), the difference between the function and a given Taylor polynomial grows. Such behavior is illustrated in Figure 9.44 for the case of f ) sin epanded about π.